Thursday, April 02, 2009

Functional Analysis for Quantum Mechanics

Noted here if and when I take a course on functional analysis. Proximate reason - I am today revising book one of SM358, the Open University's course on QM and I was just checking that position eigenfunctions were in fact delta functions.

Still at the point of scratching my head a little for a rigorous derivation as to why this is the case, although the OU text has a perfectly OK hand-wavy and intuitive motivation using approximations to delta functions they call 'top-hat functions'.________

Roy Simpson writes:

"Yes Reed and Simon is the classic text on this topic. However I think that it will be a major study course itself to go into this, partly because Functional Analysis has to deal with the many infinite structures in QM. For example the Hilbert space is infinite dimensional, so what are the definedness and convergence properties of the operators (like the Hamiltonians you will be generating) on infinite dimensional vector spaces? You will be glad just to write down a correct Hamiltonian (for the square well, etc), never mind proving that it has the correct infinitary properties! The Dirac Delta "function" is the simplest of another kind of mathematical entity known as a Distribution. Distributions D are dual to functions: D: Fn --> Real -- (where Fn is: Real --> Real). My short book on Distributions is by Schwartz and is quite readable (on a second attempt!). It all relates to complex numbers too. Looking at Wikipedia I see that the Dirac delta can be formed into a "comb" and used for Digital Signal Processing."